The problem of \(L^q\) -maximal regularity for (REF ) has been raised by P.-L. Lions a decade ago in a series of seminars, and, roughly speaking, is a semilinear version of the classical Calderón-Zygmund linear maximal regularity; that is, under the assumption that \(f\)  is bounded in \(L^q\) , then one should be able to have a control of \(H(Du)\)  and \( \operatorname{tr}\left(A D^2 u\right)\)  in \(L^q\) . Lions conjectured that this should be possible provided that \(q > q_0\)  (\(q_0\)  as in (REF )), and the conjecture has been shown to be true in the recent work [1]}. The results in [1]} still have some limitations: they hold for classical solutions, \(q > 2\)  is required (while it may happen that \(q_0 < 2\)  for some \(N, \gamma \) ) and they are not local but require \(f(x)\)  to be periodic in \(x\) . Moreover, no \(x\)  dependence is allowed in first and second order terms.
